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Hi, to convert a Mesh to a body, you need to first disable "Capture Design History". You can do this if you select the top element of the object tree (filename component) and then right click. There's an option "do not capture Design History".
Once this is done you can use the function "Mesh to BRep" that creates a solid from your mesh (and then you can apply an appearance for the Fabric). Just right click on each mesh body and you can see the menu option.
For the box there is no issue, but usually when there are more than 10.000 to 15.000 triangles the application complaints and more than this it doesn't even let you convert it. No issue for the box example but it could be an issue for other projects you have (there are several mesh tools and you can also try to reduce mesh to lower the number of triangles).
Once you have the objects, in your case they are converted to surfaces (if it was a closed object it would be converted to a solid). I noticed in your box example the faces are inverted (normal is looking inside) so you need to reverse normal for each face. This can be done with the Patch workspace (command Modify-Reverse normal). Once everything looks fine you can then apply appearance to the bodies. If you want to have fabric appearance also inside then you need to convert these surfaces to solid. I would combine them to make one single body and then Create-Thicken to your desired thickness.
Regarding the smoothing, before converting to BRep you can edit a mesh and apply several operations. As stated before you can reduce mesh to have less triangles, you can remesh and more important for you, you can smooth the mesh. So before anything apply first this smoothing operation.
I have attached the box example converted to surfaces, with faces normals reversed and fabric material applied.
If this helps you please like it, and if it works as a solution please mark it accordingly. Helps me stay engaged with the forum 
In one word. No!
What @norbertut has suggested is incorrect as when converting a triangulated mesh directly to a BRep :
1. in this case will not result in a solid but surfaces
2. will translate exactly the faceted nature of the mesh and will not be smooth.
Having said that, this object should not be so hard to re-create with a T-Spline in Fusion 360.
Is this close enough ?
@norbertut wrote:
Hi, @TrippyLighting, I think you went through my reply quite fast...
For your first point, I stated:
"Once you have the objects, in your case they are converted to surfaces (if it was a closed object it would be converted to a solid)."
For second point, I was also suggesting to smooth the mesh before converting to BRep.
I agree with you it is difficult to modify an OBJ and it is often better to draw from scratch, but the box was just an example of a workflow @Anonymous is trying to put in place.
1. That is probably true 😉
2. How do you propose to "smooth" a triangulated mesh ?
I will continue to refer to "triangulated mesh" instead off a particular file extension.
The .obj format is perfectly fine to transport high quality quad meshes, textures (not used by Fusion 360) etc.
@Anonymous you workflow needs an update if your other software can only export triangulated garbage.
@norbertut the entire purpose of re-meshing is to create quad-mesh topology. That will import and convert fine into a T-Spline in Fusion 360. I use this process all the time and have demonstrated numerous times how that can be done.
I could have done the same with Instant Meshes. The matter is, however, all of the re-meshing methods will require some form of manual work afterwards.
I am still wondering what other software you are using that creates only triangulated meshes as output.
@Anonymous wrote:
Although my problem is solved, I have to leave my original question as unsolvable because Fusion 360 does not have an easy solution to do what I originally wanted.
You can mark this as solved. There is no easy solution for converting triangulated meshes into quad meshes.
This is a mathematical reality in any 3D software, whether it's a Sub-D modeling, NURBS modeling or CAD software.
I've done this stuff for almost 30 years and you can safely assume that I have a pretty good understanding of the matter.
Quad meshes when used as control point meshes as is the case in Sub-D modeling, T-Spline and PolyNURBS modeling have a topology and with T-Splines and PolyNURBS being able to create resolution free, mathematically precise surfaces.
The mathematical precision is lost when a surface is triangulated into a fixed resolution mesh and the object looses its topology. In essence there is a significant loss of information when going from a NURBS model to a triangulated object. That information cannot be fully re-created even with re-meshing and other reverse engineering software. It can only be approximated.
Complex math does not have easy buttons!
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